Can't bevel - Invalid boundary region to join faces - edge dissolve
Got the answer thanks to advice on: Can&#39;t bevel an NGon

I learned that blender 'needs' the extra edges, and that having an ngon is a bad idea. So I created all the extra geometry (by extruding along normals and manipulating some vertices/faces), and then the bevel works:

(I also now created the whole mesh without doing a boolean operation).